Derivation of a Mathematical Model for the Growth of Anodic TiO₂Nanotubes under Constant Current Conditions

Abstract

Due to certain limitations of traditional models, the growth mechanism of porous anodic TiO 2 nanotubes has not been well determined currently. Herein, for the first time, a mathematical model of voltage-time transient curves under constant current conditions is derived theoretically, based on the conception of ionic and electronic currents and Ohm's law. The simulation results show high fidelity to the experimental curves, and illustrate the linear correlation between nanotube length and ionic current. Further, based on this model, the avalanche breakdown can be explained, which shows advantage over the former derivations on compact films. And this model indicates that the discrepancy between compact and porous oxide films lies on the magnitude of electronic current during anodization. Moreover, the proportion of the ionic and electronic current is then calculated during constant current anodization. It can be concluded that the ionic current contributes to the oxide growth while the electronic current gives rise to the oxygen bubble evolution which acts as the growth mould of the oxide. The present results promote the understanding of the growth kinetics of porous anodic oxides from qualitative interpretation to quantitative analyses.

Evidence of oxygen bubbles forming nanotube embryos in porous anodic oxides

“…Open However, based on the ionic current and electronic current theories 31,52,[54][55][56][57][58] and the 'oxygen bubble model', 24,[59][60][61][62] three typical stages in Fig. 1d can be given a clear explanation.…”

“…According to the ionic current and electronic current theories, 31,52,[54][55][56][57][58] the formation of anodic TiO 2 nanotubes is a result of the oxide flow around the oxygen bubble. Zhou et al 20 suggested that when the growth of oxide exceeded the rate of the evolution of oxygen bubbles, the viscous flow of oxide around the oxygen bubbles led to the formation of the nanotube embryos.…”

“…30 The relationship between the field-assisted dissolution reaction of fluoride ion and the total anodizing current cannot be explained up to now. [30][31][32] Thompson et al 9 clearly pointed out that the above field-assisted dissolution reaction is pure chemical reaction, independent of the applied electric field. 9 In summary, the limitation of the above field-assisted dissolution reaction is that the physical meaning of electric field is not clear.…”

Debunking the essential effect of temperature and voltage on the current curve and the nanotube morphology

“…A new explanation for the decreasing current under low temperature is given based on the 'oxygen bubble model' [32][33][34][35][36][37][38] and the ionic current and electronic current theories. [39][40][41][42][43][44][45] We propose that the electronic current, rather than the eld-assisted dissolution, plays an important role in the growth of nanotubes. During the anodization, the electronic current leads to the formation of oxygen bubbles.…”

Light-Confining Nanoporous Anodic Alumina Microcavities by Apodized Stepwise Pulse Anodization

“…Furthermore, anodization of aluminum is a well-established electrochemical process performed in mild acid electrolyte solutions that is economical and fully scalable process and requires minimum safety measures. [30][31][32][33][34][35][36] Recent studies have demonstrated that the effective refractive index of NAA can be precisely modulated in a multi-dimensional fashion to create a broad range of PC structures with finely tuned optical properties (e.g. distributed Bragg reflectors, gradient-index filters, bandpass and linear variable bandpass filters, encoded photonic tags, etc.).…”